The latest version of the iPhone OS allows you to have 11 pages of 16 icons, plus the four permanent icons at the bottom of the screen, for a total of 180 apps. You can actually have more than 180, though only 180 icons will be visible.

To get to the invisible apps, just do a search (press Home from the home screen), and type part of the non-visible app's name. I don't know how many of these invisible apps you can have, as I have not tried more than three so far.

[robg adds: As far as I know, the limit is simply based on the available space on your device. Long before that point, though, you may run into a limit with your ability to remember the names of all the invisible apps.]

There is only the limitation of how much space you have left on your iPhone or iTouch. I have over 300 apps with no problem. Granted, it can be confusing to remember what is where. Weather, YouTube, Notes, Maps, and other TECH oriented apps I use frequently I know by name are set aside and I access solely through the 'Spotlight' search ...; Now, if they would only allow 5 across ... that would help a LOT!!! They do it in Jailbroken but I choose not to take that route.

Instead of trying to remember the names of all the apps (I currently have 224), every time I purchase/get a new app, I write the name in Notes (Home page). The list will soon be a long one, but at least you'll know what you have.